Page 190 - Linear Algebra Done Right

The Minimal Polynomial
The Minimal Polynomial
As we will soon see, given an operator on a finite-dimensional vec- 179
tor space, there is a unique monic polynomial of smallest degree that A monic polynomial is
when applied to the operator gives 0. This polynomial is called the a polynomial whose
minimal polynomial of the operator and is the focus of attention in highest degree
this section. coefficient equals 1.
Suppose T ∈L(V), where dim V = n. Then For example,
2
8
2 + 3z + z is a monic
2
2
(I, T, T ,...,T n )
polynomial.
cannot be linearly independent in L(V) because L(V) has dimension n 2
2
(see 3.20) and we have n +1 operators. Let m be the smallest positive
integer such that
m
2
8.33 (I,T,T ,...,T )
is linearly dependent. The linear dependence lemma (2.4) implies that
one of the operators in the list above is a linear combination of the
previous ones. Because m was chosen to be the smallest positive in-
teger such that 8.33 is linearly dependent, we conclude that T m is
2
a linear combination of (I,T,T ,...,T m−1 ). Thus there exist scalars
a 0 ,a 1 ,a 2 ,...,a m−1 ∈ F such that
2
a 0 I + a 1 T + a 2 T +· · ·+ a m−1 T m−1 + T m = 0.
The choice of scalars a 0 ,a 1 ,a 2 ,...,a m−1 ∈ F above is unique because
two different such choices would contradict our choice of m (subtract-
ing two different equations of the form above, we would have a linearly
dependent list shorter than 8.33). The polynomial

2
a 0 + a 1 z + a 2 z +· · ·+ a m−1 z m−1 + z m
is called the minimal polynomial of T. It is the monic polynomial
p ∈P(F) of smallest degree such that p(T) = 0.
For example, the minimal polynomial of the identity operator I is
2
z − 1. The minimal polynomial of the operator on F whose matrix

2
equals 41 is 20 − 9z + z , as you should verify.
05
Clearly the degree of the minimal polynomial of each operator on V
2
is at most (dim V) . The Cayley-Hamilton theorem (8.20) tells us that
if V is a complex vector space, then the minimal polynomial of each
operator on V has degree at most dim V. This remarkable improvement
also holds on real vector spaces, as we will see in the next chapter.